III.

Euclidean Geometry

Perhaps the most familiar and intuitive geometry is called Euclidean geometry. Euclidean geometry describes most aspects of the everyday world and was named after Euclid, the ancient Greek mathematician who developed it. While the postulates of Euclidean geometry do seem plausible when applied to physical space in our universe, there is evidence that Euclidean geometry is not the perfect system for describing space.

Two-dimensional Euclidean geometry is often called plane geometry; three-dimensional Euclidean geometry is frequently referred to as solid geometry. Plane geometry deals with figures that lie wholly in one plane. A plane may be measured in terms of two dimensions: length and width. Solid geometry deals with figures that have three dimensions: length, width, and height.

Conic sections, a commonly studied topic of geometry, are two-dimensional curves created by slicing a plane through a three-dimensional hollow cone.

A.

Euclid’s Postulates

Euclid, who lived about 300 bc, realized that only a small number of postulates underlay the various geometric theorems known at the time. He determined that these theorems could be deduced from just five postulates.

1. A straight line may be drawn through any two given points.

2. A straight line may be drawn infinitely or be limited at any point.

3. A circle may be drawn using any given point as the center, and with any given radius (the distance from the center to any point on the circle).

4. All right angles are congruent. (A right angle is an angle that measures 90°. Two geometric figures are congruent if they can be moved or rotated so that they exactly overlap.)

5. Given a straight line and a point that does not lie on the line, one and only one straight line may be drawn that is parallel to the first line and passes through the point.

These five postulates can be used in combination with various defined terms to prove the properties of two- and three-dimensional figures, such as areas and circumferences. These properties can in turn be used to prove more complex geometric theorems.

B.

Two-Dimensional Euclidean Figures

Figures commonly encountered in two-dimensional geometry include circles, polygons, triangles, and quadrilaterals. Triangles are actually three-sided polygons; quadrilaterals are polygons with four sides.

B.1.

Circles

A circle is a plane curve where all points are equidistant from a point in the plane called the center. Only one circle may be drawn that passes through three noncollinear points. The word circle is sometimes used to mean the entire portion of the plane enclosed by the curve rather than just the points that lie on the curve.

Concentric circles are circles that have a common center. An angle is called a central angle of a circle if its vertex (point where the two arms of the angle meet) is at the center and its sides are radii of the circle. The circumference of a circle is divided into 360 equal degrees, and the number of degrees in a central angle is equal to the number of degrees in the intercepted arc on the circle.

The area of a circle is equal to the product of the circumference and the diameter divided by 4, or A = Cd/4. The ratio of the circumference to the diameter is approximately 3.14159265. This constant number, called pi (p), has an infinite number of nonrepeating digits. The area of a circle may also be written A = pr², where r is the radius. Similarly, the circumference is equal to the product of the diameter and the constant pi: C = pd

B.2.

Polygons

Any plane figure bounded by straight lines is a polygon. If all of a polygon’s sides are of equal length and the angles are also equal, the figure is a regular polygon. The apothem of a regular polygon is the distance from the center of the polygon to a side. The area of a regular polygon is equal to the product of one half the apothem and the perimeter, or A = ½ap:

B.3.

Triangles

A triangle is a plane figure bounded by three straight lines. A scalene triangle has three sides of unequal lengths, an isosceles triangle has two equal sides, and an equilateral triangle has three equal sides:

In the isosceles triangle the angles opposite the equal sides are equal, and in an equilateral triangle all three angles are equal.

A right triangle is a triangle in which one angle is a right angle. The side opposite the right angle is called the hypotenuse; the two adjacent sides, the legs. The famous Pythagorean theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs, or c² = a² + b².

The angles inside a triangle are called the interior angles; those formed by extending a side of the triangle, exterior angles. The sum of the interior angles of any triangle equals 180°. Also, an exterior angle is equal to the sum of the remote interior angles (the two interior angles that do not share a side with the exterior angle): ÐD = ÐA+ÐB.

A line drawn from a vertex of a triangle to the midpoint of the opposite side is called a median. The three medians of a triangle meet at a point two-thirds of the distance from the vertex to the midpoint of the opposite side. An altitude of a triangle is the length of the line connecting a vertex and the side opposite that vertex that is also perpendicular to the opposite side. (Two lines are perpendicular if they meet in a right angle.)

Two triangles are congruent if they satisfy any of the three following sets of conditions: (1) two angles and a side of one triangle are equal to the corresponding side and two angles of the other triangle; (2) two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle; or (3) three sides of one triangle are equal to three sides of the other triangle. If the triangles can be perfectly overlapped without removing either from their common plane, they are directly congruent; if one must be flipped over, they are inversely congruent.

If two triangles have equal angles, the triangles are said to be similar and the corresponding sides are in proportion to one another.

The area of any triangle is equal to the product of one-half of a base and the altitude perpendicular to that base: A = ybh. (Any side can be considered the base of a triangle, but usually the side on the bottom is so designated.) If the triangle is equilateral, the area is given by , where a is the length of the side. If the sides of any triangle are a, b, and c, the area is given by a relation credited to the ancient Greek mathematician Archimedes: where s is half the perimeter (s = ya + yb + yc).

B.4.

Quadrilaterals

A quadrilateral is a plane figure bounded by four straight lines. There are several familiar types of quadrilaterals. Trapezoids are quadrilaterals that have two parallel sides of unequal lengths. Parallelograms are quadrilaterals that have opposite sides of equal length. A rhombus is a parallelogram (and therefore also a quadrilateral) whose sides are equal, a rectangle is a parallelogram whose angles are all right angles, and a square is a parallelogram whose angles are right angles and whose sides are of equal length. The diagonals of a parallelogram bisect each other; if the parallelogram is a rectangle, the diagonals are also equal. Irregular quadrilaterals have four unequal and nonparallel sides:

The area of a trapezoid is half the sum of the bases times the altitude, or A = [(b₁ + b₂)/2]h. For a parallelogram, area equals base times height: A = bh.

For irregular quadrilaterals, a good method for determining the area is to divide the figure into two triangles by means of a diagonal, then find the individual areas of the triangles and add them together.

C.

Three-Dimensional Euclidean Figures

Figures commonly encountered in three-dimensional geometry include spheres, polyhedrons, prisms, pyramids, cylinders, and cones. Cylinders are actually special cases of prisms; cones are special cases of pyramids.

C.1.

Spheres

A sphere is a surface where all points are equidistant from one point, called the center. If a plane cuts a sphere, the points where they intersect form a circle. The largest circle (called a great circle) is produced when the plane passes through the center of the sphere.

The equator on Earth is a great circle. The surface area of a sphere is given by A = 4pr², its volume by V = 4/3pr³.

C.2.

Polyhedrons

A polyhedron is a figure bounded by plane surfaces. If the faces of the polyhedron are all congruent regular polygons, the polyhedron is said to be regular. It has been proven that the five regular polyhedrons—the tetrahedron (four sides), cube (six sides), octahedron (eight sides), dodecahedron (12 sides), and icosahedron (20 sides)—are the only ones possible. These five polyhedrons were known to the ancient Greek geometers. All polyhedrons (regular or not) have the remarkable property that the number of faces (the flat sides) plus the number of vertices (the angles where edges intersect) equals the number of edges plus 2. Up to relatively recent times, polyhedrons were believed to have mystic associations with natural phenomena.

C.3.

Prisms

A prism is a polyhedron that has parallel and congruent polygons, called bases, for two faces and parallelograms for all other faces. A parallelepiped is a variety of prism whose bases are parallelograms. A right prism has rectangles for sides (but not necessarily for bases). The volume of any prism is equal to the area of one of its bases times its height: V = bh.

C.4.

Pyramids

A pyramid is a polyhedron that has a polygon as its base and sides that consist of triangles having a common vertex, called the apex. A pyramid is a regular right pyramid if its base is a regular polygon and if a line joining the center of its base to its apex is perpendicular to its base. The volume of any pyramid is equal to one-third the area of its base times its height: V = €bh.

C.5.

Cylinders and Cones

A cylinder is a prism with circular bases. The formula for the volume of a cylinder is therefore the same as for a prism: A = bh. If the line connecting the centers of the two bases is perpendicular to those bases, the cylinder is a right cylinder; otherwise, it is oblique.

A cone is a pyramid with a circular base. A cone is a right cone if a line joining the center of its base to its apex is perpendicular to its base. The formula for the volume of a cone is the same as for a pyramid: V = €bh.

D.

Conic Sections

Conic sections are curves formed by the intersection of a plane with the surface of a cone. (When discussing conic sections, cone means two right circular cones placed apex to apex.) The surface of the cone on either side of the apex is called a nappe of the cone. If A is the angle between the axis of the cone and its surface and the cone is cut by a plane that makes an angle with the axis that is greater than A, the intersection is a closed curve called an ellipse. If the plane and the axis are perpendicular, the intersection is a circle, which is considered a special case of the ellipse.

If the plane intersects the axis at an angle equal to A, so that the plane is parallel to the surface of the cone, the intersection is an open curve of infinite extent called a parabola. If the cone is intersected by a plane that is either parallel to the axis or makes an angle with it that is smaller than A, and if the plane does not contain the apex of the cone, the intersection is called a hyperbola. In this case the cone is necessarily intersected in both nappes, and it follows that the hyperbola has two branches, each of which is infinite in extent.

Conic sections are two-dimensional or plane curves, and therefore a desirable definition of conic sections avoids the notion of a cone, which is three-dimensional. A conic section may be two-dimensionally defined as the set of points of which the distances from some fixed point are in a constant ratio to the distances of the points from a fixed line that does not pass through the fixed point. The fixed point is called the focus, and the fixed line is called the directrix. The constant ratio is called the eccentricity of the conic section and is usually denoted by the letter e. If P is a point and Q is the foot of a line from P perpendicular to the directrix, the point P is on the conic section if and only if [FP] = e[QP], in which [FP] and [QP] are the distances between the respective points. When e = 1, the conic section is a parabola; when e > 1, it is a hyperbola; and when e < 1, it is an ellipse.

The conic sections have numerous mathematical properties that give them important applications in mathematical physics. For example, light reflected by mirrors molded to the curve of a conic section has particular characteristics: Rays emanating in any direction from the center of a circle are reflected back to the center; rays emanating in any direction from one of the two foci (geometrical centers) of an ellipse are reflected to the other focus. Parabolic mirrors are often used in spotlights because the rays emanating from the focus of a parabola are reflected out in parallel lines, minimizing spread:

Rays emanating from one focus of a hyperbola are reflected in such a direction that they appear to emanate from the other focus.